Orthogonal Polynomials and Special Functions

October 1998 Orthogonal Polynomials and Special Functions Newsletter 6
Paco Marcellán and Renato Álvarez-Nodarse did a superb
job in making everyone comfortable. Renato’s boundless
energy, enthusiasm and friendliness, and Paco’s leadership
are an excellent recipe for a great conference.
Most of the talks, both invited and contributed, were
first-rate. It was remarkable to see the breadth and depth
of the Spanish school of orthogonal polynomials and special
functions, many of whose members are academic descendants
of Paco Marcellán.
Another unique feature of this meeting was its combining
numeric and symbolic computations. The older tradition
of numeric computation, and the younger tradition of
symbolic computations have completely different cultures,
concerns, and methodology. I am sure that they both can
benefit from a cross-fertilization. By choosing three of the
invited speakers (Golub, Gautschi, Maday) from the former
camp, and the other three (Koepf, Petkovˇsek, Zeilberger)
to be from the later, the participants and speakers
each learned much more than they would in yet another
specialized meeting in their field.
I myself, who is almost ignorant of numerical analysis,
learned so much from Gautschi’s and Golub’s talk, and
even from Maday’s 300 words per minute talk.
The conference also fostered many personal discussions.
In particular, I had a fascinating discussion with Marcel de
Bruin (who was the only one, incidentally, to hand in the
homework that I have assigned during my talk), about determinants.
He turned out to be a real determinant-whiz,
and later has sent me little-known, but very interesting,
papers by van der Corput (in Dutch!).
On a more personal note, I was happy to meet Gene
Golub for the first time, after I have heard so much about
him from our mutual friend Marvin Knopp. I was also
happy to meet the famous Walter Gautschi, who was the
first to believe de Branges!
As the years go on, the memories of most conferences
blend into a dull continuum. Not this one! It will always
be remembered as a very happy singularity.
Doron Zeilberger
(zeilberg@euclid.math.temple.edu)
3. Minisymposium on Problems and Solutions in
Special Functions
SIAM Annual Meeting 1998: Toronto, Canada,
July 13-17, 1998
On July 14, 1998, our Activity Group sponsored a Minisymposium
Problems and solutions in Special Functions
(Organizers: Willard Miller, Jr. and Martin E. Muldoon)
at the SIAM Annual Meeting in Toronto. The organizers
recognized that by providing concrete and significant problems,
the problem sections in journals such as SIAM Review
and the American Mathematical Monthly have been
influential in advancing mathematical research and have
played a role in attracting young people to the mathematical
profession. At a time when the SIAM Review is phasing
out its problem sections (see Newsletter 8.2, pp. 19–21)
it seemed appropriate to assess the history and impact of
the problems sections and their future evolution.
Cecil C. Rousseau, University of Memphis offered a retrospective
on the 40-year history of the SIAM Review
Problems and Solutions Section, based on his experience as
a collaborating editor and then as an editor of the Section.
We learned that of the 777 problems proposed, 329 were
starred (no solution submitted by the proposer). The title
most used was “A definite integral” while the keywords
occurring most frequently were “integral” (131 times), “inequality”
(47), “identity” (33), “series” (25) and “determinant”
(24). The most frequent problem proposers were M.
S. Klamkin (46), M. L. Glasser (38), D. J. Newman (24)
and L. A. Shepp (20).
Cecil chose a specific issue (April, 1972) and mentioned
Problem 72-6 by Paul Erdős (A solved and unsolved graph
coloring problem) that provided the first contact between
Erdős and the Memphis graph theory group (Faudree, Ordman,
Rousseau, Schelp), and in that way led to more than
40 joint papers involving Erdős and the members of this
group. He mentioned Problem 72-9 (An extremum problem)
by Richard Tapia, who, coincidentally, was honored
on the same day as our Minisymposium by a Minisymposium
for his 60th birthday. In the same issue, the solution
to Problem 71-7 (Special subsets of a finite group) was
the very first publication by Doron Zeilberger. Rousseau
himself had a solution of Problem 71-13 proposed by L.
Carlitz, which called for a proof that a certain integral
involving the product of Hermite polynomials was nonnegative.
At the time, Rousseau looked for, but did not find,
a combinatorial interpretation of the integral that would
immediately imply its nonnegativity. That there is such an
interpretation was shown by Foata and Zeilberger in 1988.
Later in the discussion, Rousseau mentioned that problems
sometimes get repeated in spite of the best efforts of the
editors; for example, Problem 95-6 repeats part of Problem
75-12 but that he had found the relevant double integral
later in Williamson’s Calculus (6th ed), 1891!
Otto G. Ruehr, Michigan Technological University, discussed
the forty-year history of the Section with particular
attention to the second half. He offered an anecdotal
description of the trials, tribulations and satisfactions of
being editor. Special attention was paid to problems in
classical analysis, particularly those relating to orthogonal
polynomials and special functions. He regretted that
some problems he had proposed (73-12, 84-11) attracted
only one solution other than that of the proposer. Sometimes,
sheer luck played a role as in a solution of his which
depended on the relatively sharp inequality 27e 2 < 200.
In spite of the best editorial efforts, errors often crept in.